# [EM] Plurality criterion and the "sincere CW"

Toby Pereira tdp201b at yahoo.co.uk
Sun Jun 30 14:45:13 PDT 2019

``` In a previous reply on this subject, I said I'd prefer the definition to be something like:
"If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is ranked anything other than last or joint last (either explicitly or through implication on a truncated ballot), then A's probability of winning must be no less than B's."
In Woodall's definition you quote, there's no mention of an implicit approval cutoff, but maybe that's because it's implicit rather than explicit! But the point is that the definition refers to votes - "If some candidate a has strictly fewer votes in total..." and by vote it seems to means an explicit ranking of a candidate. And that wording suggests some form of approval because you are in some sense voting for that candidate. Whereas if I decide to rank every candidate, then I wouldn't consider this to be a vote for my second-to-bottom, any more than it would be if I just truncated and left the bottom two or more.
On the face of it, it does suggest that it might encourage truncation. I might hate my second-to-bottom candidate, so I wouldn't want to cast a vote for them, and if the plurality criterion says that's what I'm doing, then a method passing this criterion might punish me for it. So it seems that I should not rank any of the candidates I hate and just truncate.
Of course, it might be that in practice a method passing plurality won't punish me for a full ranking (it's not something I've greatly studied so I don't know), but the wording of the criterion itself suggests that it might.
Toby

On Sunday, 30 June 2019, 17:01:02 BST, C.Benham <cbenham at adam.com.au> wrote:

Juho (and interested others),

The Plurality criterion was coined in 1994 by Douglas Woodall. Quoting him exactly from then:

The following rather weak property was formulated with single-seat elections in mind, but it makes sense also for multi-seat elections and, again, it clearly holds for STV .

Plurality. If some candidate a has strictly fewer votes in total than some other candidate b has first-preference votes, then a should not have greater probability than b of being elected.

No mention of any "implicit approval cutoff".  I know that at the time Woodall was only thinking about strict rankings from the top with truncation allowed.
If equal-first ranking is allowed, then for the purpose of this criterion we should be using the fractional (summing to 1) interpretation of the number of
"first-preference votes".

Juho seems to think that the Plurality criterion is a "feature" or strategy device  that somehow encourages truncation.  It isn't and doesn't.

If the method uses one of the traditional Condorcet algorithms that are almost the same as each other (Smith//MinMax, Schulze, River, Ranked Pairs)
and uses Winning Votes as the measure of pairwise defeat strength, then the method meets Plurality and also has, at least in the zero-info case, a weak
random-fill incentive.

IRV, and IRV modified to meet Smith by before each elimination checking to see if there is pairwise-beats-all candidate among those remaining, both meet
the Plurality criterion.  In those methods do the voters have any have any incentive "not to rank the candidates of the competing groupings" ?  No they
don't.

So what is the point of the Plurality criterion?  To my mind it is simply about not offending obvious fairness and common-sense.

Juho, try to imagine that you have no interest in or knowledge about voting algorithms, you've never thought about the split-vote problem. You are accustomed
to voting in plurality elections (or even perhaps Approval elections) and you've never been interested in doing anything other than voting for your sincere
favourite, who regularly wins.  You are content with the current voting method and can't see any point in changing it.

Now imagine some voting-reform movement succeeds and the new method is, say, MinMax(Margins).  You hear that voters can now rank more
than one candidate and you simply seek assurance that you will be allowed to go on voting as before and you assume that the government must
more-or-less know what it's doing and assume the method won't in any way be less fair than before.

In this election your favourite is A.
46: A
44: B>C
10: C

It is announced that the winner is B.  At first you think "A got more first-preference votes than B, it must have something to do with some voters'
second preference votes", but then you notice that B got the same number of second-preference votes as A (zero), and then you ask "How on earth
did this crazy new method elect B over my favourite A, who very clearly got more "votes" (marks next to his name on the paper ballots) all of which
were first-preference votes!"

On hearing the reply "Oh, that's because B was the fewest votes shy of being the Condorcet winner" do you (a) say "Oh how silly of me, obviously
that's fair!" or  (b) say .. something much less understanding and accepting ?

This scenario also works if the old method was IRV.  You might also notice that this first MMM election scenario is also a massive egregious failure
of the Later-no-Help criterion (because if the B voters had truncated then B wouldn't have won).  Do you like that criterion?

If the old method had been Approval, you would then presumably be understanding and resigned if it is announced that C won.
In fact electing A is a failure of the Minimal Defense criterion. Do you like that one?   So methods that meet both MD and Plurality (such as Winning
Votes and Smith//implicitA) must elect C.

... methods might not elect the best winner (sincere Condorcet winner).
If voters decline to (or don't bother to) express some or all of their very weak (possibly light-minded) pairwise preferences by truncating, then I don't
classify that as "insincere" voting.  Since therefore there could be several (or even many) alternative "sincere voting" profiles it follows that there
could be more than one "sincere CW".  It seems obvious to me that the one of of those that is based on only the relatively strong pairwise preferences
will have a higher "social utility" than one based on all pairwise preferences which include a lot of very weak ones.

49: A>>>B>C
03: B>A>>>C
48: C>>>B>A

Say these are the sincere preferences. If the voters care to express all their pairwise preferences then the "sincere CW" is B, but if they choose
what I consider to be an alternative way of sincere voting and truncate where that will only "conceal" some weak pairwise preferences then
an alternative "sincere CW" is (the apparently higher Social Utility candidate) A.

In fact if the method used was the tweaked IRV  method with an explicit approval cutoff that I recently suggested and the cast votes were
49: A>>B
03: B>A>>
48: C>>B

then only C would be disqualified (because A both pairwise beats C and is more approved than C) and then B is eliminated and A wins.
I doubt that there would much blood flowing in the streets caused by the failure to elect the voted CW (B).

As consolation for not meeting the Condorcet criterion we would have a method much more resistant to Burial strategy than any Condorcet
method (and maybe more appealing to people who like IRV).

Juho Laatu juho.laatu at gmail.com
Sat Jun 29 07:43:29 PDT 2019

P.S. I don't like the plurality criterion. It actually sets an implicit approval cutoff at the end of the listed candidates. The worst part of that idea is that it encourages voters not to rank the candidates of the competing groupings. That (potentially huge amount of missing information) is not good for ranked methods. If voters learn to use that feature, methods might not elect the best winner (sincere Condorcet winner).

The following rather weak property was formulated with single-seat elections in mind, but it makes sense also for multi-seat elections and, again, it clearly holds for STV .
Plurality. If some candidate a has strictly fewer votes in total than some other candidate b has first-preference votes, then a should not have greater probability than b of being elected.

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